What are we doing when we do mathematics?

[epicurus]

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I am a mathematician turned cognitive scientist/​AI researcher. I wrote a short essay on the philosophy of mathematics that I think will be of interest to many people here. I combine platonist and formalist ideas and frame mathematics as an experimental activity, extremely similar to physics and other hard sciences with computation playing a parallel role to experimentation, and the creation of definitions and axiomatic frameworks being similar to the creation of physical theories like Newtonian gravity or Quantum Mechanics.

This perspective also stresses the non-formalistic aspects of mathematical research by gesturing at a strong division of intellectual activity into semantics and syntactics. Although this is by far the least developed part of the essay, it is the secret motivation behind me writing the essay in the first place, and my core interest. I am secretly using mathematics as a test case to probe the nature of general cognition.

[TAG]

The second view [formalism] grew in strength as a reaction to the ”foundational cri-sis” at the turn of the twentieth century and is a more mechanical view of mathematics. It is the attitude that we present to students, especially when we ask how mathematics is to be certified correct.

You didn’t say really what formalism is.

Formalism a a form of anti realism. Formalists do not regard mathematical. objects as existing mateeial.objects or as existing abstract objects. Formalism is of course *also* a positive claim about mathematical truth...that it is exploring the consequences of a particular formal game.,.and is therefore relative to a game. And it recommends mechanical procedures for checking and generating proofs, although how much that is possible is in doubt. So, for formalists there is no answer or meaning to foundational question like “are these the true axioms and rules of inference”.

The first view [Platonism] is intentionally vague

It’s defined as

“the view that mathematical objects (like numbers and sets) exist independently of human minds and language, and are real, abstract entities”.

and centers mathematics within the broader domain of the sciences—mathematical objects are ”real”, and mathematicians study them in similar ways to how physicists might study electrons.

Platonism is a form of realism, but not a form of realism about ordinary material objects … the definition makes that clear. Therefore ,itisnitiis not something that is very analogous to scientific realism. For Platonists , there is an ultimate truth, to the foundational questions, but there is no mechanical. procedure for converging on it...it comes down to intuition, and intuitions vary. Platonic realism, as a source of not just proof, but just truth , needs a way manifest in the procedures of mathematicians. Are their physical brains causally interacting with immaterial entities? If the de facto standard of truth is derivation from axioms, fomralism is good enough, and Platonism falls to occams razor.

But anyway...Platonism is realism and formalism is anti realism. So how can they be reconciled?

I propose that these views can be reconciled by understanding mathemat-ics as a cyclic process that moves between computation and theory-building.

This cycle proceeds as follows:

1. We perform computations—physical processes that generate observa-tions and constitute mathematical experiments.

1. Based on these observations, we build theories (definitions, axioms, rules) to explain what we’ve observed.

1. Within these theories, we make new computations that both test our theories and generate new observations.

1. These observations lead to theory refinement, beginning the cycle anew.

To run a computer programme is to run a formalization of a problem. Computation Is formal not Platonic, in the sense that there is no way for an immaterial Platonic entity to make the computer swerve from its physically determined course . The very idea is even less likely than that of Platonic entities influencing mathematicians brains. Remember, Platonism isn’t the claim that maths is about physical entities.

Is it an experiment? Up to a point. On the one hand, you are looking at the result. On the other hand, you could, in principle have figured out that result , and are just using the computer as an extension to your brain...the result is apriori, and so.not genuinely empirical.

Similarly, a mathematical object ”feels real” to us as we interact with it in a variety of ways that do not lead to contradictions.

In this case, interacting with a mathematical object is a purely computational activity but no less ”real” for that.

Do we literally interact with mathematical objects ?

Are feelings necessarily so?

[Knight Lee]

The difference between mathematicians and physicists, is that physicists are studying a mathematical structure while living inside it.

• See also: Mathematical universe hypothesis